Existence of Traveling Wave Solutions to the Problem of
Soil Freezing Described by a Model Called M1
YOSHISUKE NAKANO
INTRODUCTION
The scientific study of soil freezing and ice segregation began in the early 1900s. By the
1930s researchers (Taber 1930, Beskow 1935) had already found that ice segregation and
the resultant frost heave are caused not only by freezing of in-situ water, but also by freez-
ing of water transported toward a freezing front from the unfrozen part of the soil. The
understanding gained in the 1930s was largely qualitative. However, the transport of
water was already identified as one of major issues in the study of soil freezing. The prob-
lem has attracted the attention of many researchers (see Nakano 1991).
The main constituents of saturated, frozen, and fine-grained soils are a solid porous
matrix of soil particles and ice, and water in the liquid phase called unfrozen water. The
physical properties of all constituents except unfrozen water are well understood. It is gen-
erally understood that the transport of water in frozen soils is mainly caused by the move-
ment of unfrozen water and that unfrozen water exists in small spaces surrounded with
surfaces of soil particles and ice. Heaving during freezing is not limited to water in soil
systems. It occurs with benzene or nitrobenzene in soils (Taber 1930), water in various
powder materials including hydrophobic carborundum (Horiguchi 1977), liquid helium in
porous glasses (Hiroi et al. 1989), water in hydrophobic silicon-coated glass beads (Sage
and Porebska 1993), and water in porous rocks (Miyata et al. 1994).
The dynamic and thermodynamic properties of liquids have been known to be modi-
fied by confinement in very small spaces, such as porous media, cell membranes, etc. The
problem of confined liquids has attracted the attention of researchers in many disciplines
in recent years (Granick 1991). The maximum size of confining space that significantly
modifies the property of liquid evidently depends on a kind of liquid and its confining
solid. For instance, in the case of thin quartz capillaries with sizes of the order of a micron,
the melting point of ice is practically the same as that of bulk ice (Churaev et al. 1993).
However, in much smaller capillaries of the order of 50 nm, the melting point changes.
A significant modification may occur in the dynamic behavior of water in fine porous
media. It is known (Angell 1983) that the temperature dependence of the self-diffusivity of
supercooled water can be described by a critical type of equation with a singular tempera-
ture just below the homogeneous nucleation temperature. Recently Teixeira et al. (1997)
have found that the self-diffusivity of supercooled water confined in fine porous silica cor-
responds to that of supercooled water at about 30C lower temperature. Pagliuca et al.
(1987) have shown empirically that the gradients of pressure and temperature are two in-
dependent driving forces of water flowing through various noncharged, fine porous, and
either hydrophilic or hydrophobic membranes with pore size of the order of 10500 nm at
temperatures above the bulk melting point.
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